let f1, f2 be Function of COMPLEX,COMPLEX; :: thesis: ( ( for z being Complex holds f1 . z = ((exp (<i> * z)) - (exp (- (<i> * z)))) / (2 * <i>) ) & ( for z being Complex holds f2 . z = ((exp (<i> * z)) - (exp (- (<i> * z)))) / (2 * <i>) ) implies f1 = f2 )
assume that
A2: for z being Complex holds f1 . z = ((exp (<i> * z)) - (exp (- (<i> * z)))) / (2 * <i>) and
A3: for z being Complex holds f2 . z = ((exp (<i> * z)) - (exp (- (<i> * z)))) / (2 * <i>) ; :: thesis: f1 = f2
let z be Element of COMPLEX ; :: according to FUNCT_2:def 8 :: thesis: f1 . z = f2 . z
thus f1 . z = ((exp (<i> * z)) - (exp (- (<i> * z)))) / (2 * <i>) by A2
.= f2 . z by A3 ; :: thesis: verum